c Dr Oksana Shatalov, Spring 2012
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3.9: Slopes and tangents of parametric curves
Consider a curve C given by the parametric equations
x = x(t),
y = y(t),
or in vector form:
r(t) = hx(t), y(t)i .
If both x(t) and y(t) are differentiable, then
r0 (t) = x0 (t), y 0 (t)
is a vector that is tangent to C. Its slope is:
slope =
Another way to see this is by using the Chain Rule. We have y = y(x(t)) and then
dy
dy
dy dx
dy
=
which implies
= dt .
dx
dt
dx dt
dx
dt
EXAMPLE 1. Find the equation of tangent line to the curve
x(t) = sin t,
y(t) = tan t
π
at the point corresponding to t = .
4
EXAMPLE 2. Find the equation of tangent line to the curve
x(t) = t + 1,
at the (2, 5).
y(t) = t2 + 4
c Dr Oksana Shatalov, Spring 2012
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EXAMPLE 3. Find the points on the curve
x = t + t2 ,
y = t2 − t
where the tangent lines are horizontal and there they are vertical.
EXAMPLE 4. Show that the curve
x = cos t,
has two tangents at (0, 0) and find their equations.
y = cos t sin t